Measures of Spread

Range

$\text{Range} = \text{Maximum} - \text{Minimum}$

Simple but sensitive to outliers. Only uses two values.

Interquartile Range (IQR)

$IQR = Q_3 - Q_1$

• $Q_1$ (first quartile): Median of the lower half
• $Q_3$ (third quartile): Median of the upper half
• IQR captures the middle 50% of the data
• Resistant to outliers

Identifying Outliers: 1.5 × IQR Rule

A value is an outlier if: $x < Q_1 - 1.5 \cdot IQR \quad \text{or} \quad x > Q_3 + 1.5 \cdot IQR$

Variance

The sample variance $s^2$ measures the average squared deviation from the mean:

$s^2 = \frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n - 1}$

We divide by $n-1$ (not $n$) because we use degrees of freedom to get an unbiased estimate.

Standard Deviation

$s = \sqrt{s^2} = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}}$

Interpretation: The standard deviation measures the typical distance of data values from the mean.

Properties:

• $s \geq 0$; $s = 0$ only when all values are identical
• Same units as the original data
• Not resistant to outliers
• Affected by skewness

Effect of Linear Transformations

If $y = a + bx$:

• $\bar{y} = a + b\bar{x}$ (center shifts)
• $s_y = |b| \cdot s_x$ (spread scales by $|b|$)
• Adding a constant $a$ does not change spread
• Multiplying by $b$ scales spread by $|b|$

AP Tip: Know the formula for standard deviation, and be able to interpret it in context: "The [variable] values typically differ from the mean by about [s] [units]."